What kinds of work experience is there for aspiring Paediatricians? Or where should I get work experience?

I think something else worth noting is that universities are aware of how difficult it can be to get work experience at a hospital so if you are unable to secure this, do not worry there are lots of other things you can do. What they want to see is that you are proactive and can go out there and make opportunities for yourself. It may mean gathering a list of numbers for you local GP surgeries and asking to speak to the practice manager to see if you can get some work experience.
You can also try volunteering at local youth clubs, crèche, children libraries, any outreach work that your school might do with local primary schools, tuition centres for small kids or care homes for the elderly etc. What they want to see from you is that you can work with people who are very different from yourself.
It does take a lot of patience to find these opportunities so don’t give up if your not successful. It’s all about putting yourself out there and getting outside of your comfort zone.
Best of luck.

It depends on your age. Hospitals often offer work experience placements but I think you must be over 16. Otherwise nurseries or daycares may be a place to ask for a part time job - you could get experience and get paid too.